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Jagdish C. Joshi, Walter Winter, Nayantara Gupta, How many of the observed neutrino events can be described by cosmic ray interactions in the Milky Way?, Monthly Notices of the Royal Astronomical Society, Volume 439, Issue 4, 21 April 2014, Pages 3414–3419, https://doi.org/10.1093/mnras/stu189
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Abstract
Cosmic rays diffuse through the interstellar medium and interact with matter and radiations as long as they are trapped in the Galactic magnetic field. The IceCube experiment has detected some TeV–PeV neutrino events whose origin is yet unknown. We study if all or a fraction of these events can be described by the interactions of cosmic rays with matter. We consider the average target density needed to explain them for different halo sizes and shapes, the effect of the chemical composition of the cosmic rays, the impact of the directional information of the neutrino events, and the constraints from gamma-ray bounds and their direction. We do not require knowledge of the cosmic ray escape time or injection for our approach. We find that, given all constraints, at most 0.1 of the observed neutrino events in IceCube can be described by cosmic ray interactions with matter. In addition, we demonstrate that the currently established chemical composition of the cosmic rays contradicts a peak of the neutrino spectrum at PeV energies.
Cosmic ray propagation in our Galaxy has been studied in the past several decades using many models and with increasing complexities to explain the observational results successfully. The transport equation written by Ginzburg & Syrovatskii (1964) contains various terms to include the possible gains and losses in the flux of cosmic rays. The simple leaky box model and its variants were widely used to explain the observed secondary-to-primary GeV cosmic ray flux ratios (Shapiro & Silberberg 1970; Cowsik & Wilson 1973). Cosmic rays diffuse through the Galaxy, interact with matter and background radiations producing secondary particles of lower atomic numbers (Z). More complex models of cosmic ray propagation including the effects of energy-dependent diffusion coefficient (D) and re-acceleration were subsequently introduced by Gupta & Webber (1989), Gaisser (1991), Berezinskii et al. (1990), and Letaw, Silberberg & Tsao (1993).
In this work, we consider the steady state flux of cosmic rays for the calculation of the diffuse neutrino flux produced in cosmic ray interactions, directly based on cosmic ray observations. Thus, our results neither depend on the unknown injection spectrum nor on the escape time of very high energy cosmic rays (VHECRs).
The detection of very high energy and ultrahigh energy cosmic rays by air shower experiments (Chou et al. 2005; Risse et al. 2005; Abbasi et al. 2010; Apel et al. 2013; Knurenko & Sabourov 2013; The Pierre Auger Collaboration 2013) have an enormous impact on our understanding of the high-energy phenomena in the Universe. The compilation of cosmic ray data from various air shower experiments show a knee region near 3 PeV and ankle region near 104 PeV in the all-particle cosmic ray spectrum (Gaisser, Stanev & Tilav 2013).
If we consider the propagation of these cosmic rays within the Galaxy, secondary gamma rays and neutrinos will be produced by their interactions with Galactic matter (Evoli, Grasso & Maccione 2007; Gupta 2012, 2013; Stecker 2013). The IceCube experiment has detected some neutrino events in TeV–PeV energies which are unlikely to be of atmospheric origin (Aartsen et al. 2013; IceCube Collaboration 2013). The implication of the IceCube neutrinos for cosmic ray transition models has been studied in Anchordoqui et al. (2013), assuming that these could be of Galactic origin. Cosmic ray interactions in the inner Galaxy have been considered as the possible origin of the some of the IceCube-detected events and Fermi/Large Area Telescope (LAT)-observed gamma rays in Neronov, Semikoz & Tchernin (2013). The five shower-like events correlated with the Galactic Centre region (Razzaque 2013) could have originated from cosmic ray accelerations in supernova remnants. The correlation of the gamma-ray and the neutrino fluxes and the Galactic origin of the IceCube events have been studied in Ahlers & Murase (2013). They point out that within wide angular uncertainties of the Galactic plane, it is plausible that about 10 events are of Galactic origin. Recently, the sub-PeV and PeV neutrinos have been correlated with the cosmic rays above the second knee in the VHECR spectrum, for various sources within hadronic interaction (Murase, Ahlers & Lacki 2013) and hypernova remnants have been suggested as their sources (Liu et al. 2013). The neutrino events discovered by IceCube can also come from pγ interactions, as it is, for instance, discussed by Winter (2013), Murase & Ioka (2013), and Stecker (2013).
In this work, we study if the TeV–PeV neutrino events detected by IceCube above the atmospheric background have originated from interactions of VHECRs with Galactic matter or gas. The interactions of cosmic rays with Galactic matter also lead to the production of high-energy gamma-rays which contribute to the background measured by Fermi/LAT.
1 PROTON INTERACTIONS AND TARGET GEOMETRY
VHECRs interacting with Galactic matter give charged and neutral pions. The charged pions decay to muons and muon-type neutrinos (|$\pi ^{\pm }\rightarrow \mu ^{\pm }+\nu _{\mu } (\bar{\nu }_{\mu })$|. The muons subsequently decay to electrons, electron-type neutrinos and muon-type neutrinos |$(\mu ^{\pm }\rightarrow {\rm e}^{\pm }+\nu _{{\rm e}}(\bar{\nu }_{\rm e})+\bar{\nu }_{\mu }(\nu _{\mu })$|. The ratio of the neutrino fluxes of different flavours produced in this way is |$\nu _{\rm e}+\bar{\nu }_{\rm e}:\nu _{\mu }+\bar{\nu }_{\mu }:\nu _{\tau }+\bar{\nu }_{\tau }$| = 1:2:0.
The fluxes of neutrinos of each flavour are expected to be roughly equal on Earth after flavour mixing |$\nu _{\rm e}+\bar{\nu }_{\rm e}:\nu _{\mu }+\bar{\nu }_{\mu }:\nu _{\tau }+\bar{\nu }_{\tau } \simeq$| 1:1:1 (Gaisser 1991). For the numerical calculations, however, we compute the flavour mixing precisely using the current best-fitting values from (Gonzalez-Garcia et al. 2012, first octant solution).
For the description of the pp interactions, we follow Kelner, Aharonian & Bugayov (2006). The pp interaction time is given by tpp(Ep) = 1/(nH σpp(Ep) c), where nH is the mean hydrogen number density of Galactic matter and the cross-section of the interaction is |$\sigma _{{\rm pp}}(E_{\rm p})=34.3+1.88 \, \mathrm{ln}(E_{\rm p}/1 \, \mathrm{TeV})+0.25 \,(\mathrm{ln}(E_{\rm p}/ 1 \, \mathrm{TeV}))^2$| mb. The average (over different experiments) cosmic ray spectrum above 100 TeV from Gaisser et al. (2013) has been approximated with power laws with several breaks for our calculation; the spectrum has been linearly interpolated among (5, 0), (6.5, 0), (8.5, −0.85), (9.7, −1.7), (10.5, −1.7), (11, −2.3) on a double log scale in (log10E [GeV], log10E2.6J [GeV1.6cm−2 s−1 sr−1]).
In some models (Evoli et al. 2007), the average atomic hydrogen density in the Galaxy modelled with radii of tens of kpc and height hundreds of pc is calculated to be ∼0.5 cm−3. The density of ionized, neutral, and molecular hydrogen as a function of the height from the Galactic plane relative to the Earth's location and the radial distance from the Galactic Centre have been calculated in Feldmann, Hooper & Gnedin (2013) using the gamma-ray data observed by Fermi gamma-ray space telescope. Relative to the Earth's location, the density of atomic and molecular hydrogen gas drops from 1 to 0.1 cm−3 within a distance of 1–1.5 kpc above the Galactic plane. The density of ionized hydrogen gas steeply falls from 0.3 to 0.001 cm−3 within the same distance. The hydrogen densities of 1 cm−3 are unlikely for the tens of kpc of spherical halo as discussed in Dickey & Lockman (1990), Kalberla & Kerp (2009), and Blitz & Robishaw (2000).
It is expected to be much higher closer to the Galactic Centre. Please note that they have used a time-dependent injection spectrum proportional to |$E_p^{-2.4}$| and solved the diffusion equation to derive the steady state cosmic ray proton spectrum. We are using the observed cosmic ray spectrum in our calculations. We completely independently derive the average hydrogen density from the neutrino observations, assuming that the observed events come from interactions between cosmic rays and hydrogen within the halo. We consider different shapes of the hydrogen halo. The effective radii from equation (3) for the different geometries and the Earth 8.33 kpc off the Galactic Centre are listed in Table 1, where we denote the radius of the spherical region around the Galactic Centre by RGC.
Shape . | RGC,kpc . | hkpc . | Reff,kpc . |
---|---|---|---|
Spherical | 10.0 | 7.2 | |
Spherical | 15.0 | 13.3 | |
Cylindrical | 10.0 | 2.5 | 4.7 |
Cylindrical | 10.0 | 1.5 | 3.5 |
Cylindrical | 10.0 | 0.5 | 1.7 |
Cylindrical | 10.0 | 0.25 | 1.0 |
Cylindrical | 15.0 | 2.5 | 6.5 |
Cylindrical | 15.0 | 0.5 | 2.1 |
Cylindrical | 15.0 | 0.1 | 0.58 |
Shape . | RGC,kpc . | hkpc . | Reff,kpc . |
---|---|---|---|
Spherical | 10.0 | 7.2 | |
Spherical | 15.0 | 13.3 | |
Cylindrical | 10.0 | 2.5 | 4.7 |
Cylindrical | 10.0 | 1.5 | 3.5 |
Cylindrical | 10.0 | 0.5 | 1.7 |
Cylindrical | 10.0 | 0.25 | 1.0 |
Cylindrical | 15.0 | 2.5 | 6.5 |
Cylindrical | 15.0 | 0.5 | 2.1 |
Cylindrical | 15.0 | 0.1 | 0.58 |
Shape . | RGC,kpc . | hkpc . | Reff,kpc . |
---|---|---|---|
Spherical | 10.0 | 7.2 | |
Spherical | 15.0 | 13.3 | |
Cylindrical | 10.0 | 2.5 | 4.7 |
Cylindrical | 10.0 | 1.5 | 3.5 |
Cylindrical | 10.0 | 0.5 | 1.7 |
Cylindrical | 10.0 | 0.25 | 1.0 |
Cylindrical | 15.0 | 2.5 | 6.5 |
Cylindrical | 15.0 | 0.5 | 2.1 |
Cylindrical | 15.0 | 0.1 | 0.58 |
Shape . | RGC,kpc . | hkpc . | Reff,kpc . |
---|---|---|---|
Spherical | 10.0 | 7.2 | |
Spherical | 15.0 | 13.3 | |
Cylindrical | 10.0 | 2.5 | 4.7 |
Cylindrical | 10.0 | 1.5 | 3.5 |
Cylindrical | 10.0 | 0.5 | 1.7 |
Cylindrical | 10.0 | 0.25 | 1.0 |
Cylindrical | 15.0 | 2.5 | 6.5 |
Cylindrical | 15.0 | 0.5 | 2.1 |
Cylindrical | 15.0 | 0.1 | 0.58 |
In the following, we use Reff = 10 kpc or Reff = 1 kpc for different extreme models, but our results can be easily re-scaled with Table 1. While for the spherical halo around the Galactic Centre and extending beyond Earth Reff ∼ 7–13 kpc seems plausible, smaller values are obtained for the cylindrical haloes. For realistic scale heights, h ≲ 250 pc, Reff ≃ 1 kpc.
2 EFFECT OF COSMIC RAY COMPOSITION
The observed cosmic ray flux contains protons, helium, carbon, oxygen, iron, and heavier nuclei. In Gaisser et al. (2013), the helium nuclei flux exceeds the proton flux above 10 TeV, and at 1 PeV, helium and iron nuclei fluxes are comparable (shown with curves of different colours in fig. 4 of Gaisser et al. 2013). At 100 PeV, the cosmic ray flux contains mostly iron nuclei, and at 1 EeV, protons dominate over iron nuclei. Each nucleon in the nucleus interacts with a Galactic hydrogen atom and pions are produced, which subsequently decay to neutrinos and gamma rays. In the case of composite nucleus, the (observed) cosmic ray flux of nuclei with mass number A is JA(EA) = dNA(EA)/dEA.
Our predicted neutrino fluxes after flavour mixing for different cosmic ray compositions, nH = 1cm− 3 and Reff = 1 kpc can be found in Fig. 1. The Gaisser et al. composition has been adopted from fig. 4 in Gaisser et al. (2013), where we interpret A(EA) as a function of cosmic ray energy EA in equation (4). In that case, we linearly interpolate A between A = 4 at 5 × 104 GeV, A = 4 at 4 × 106 GeV, A = 56 at 8 × 107 GeV, and A = 1 at 109 GeV. For the ‘hypothetical model’, a helium composition between 5 × 104 and 4 × 106 GeV has been chosen, then proton between 107 and 108 GeV, and then iron at 109 GeV (and higher), linearly interpolated among these values.
First of all, since the flux roughly scales as A2−α, it is clear that the pure proton composition gives the highest flux and the pure iron composition the lowest. The Gaisser et al. model shows an iron composition at about 108 GeV, which leads to a dip in the neutrino flux at PeV energies, exactly where the excess is observed. For comparison, we show a hypothetical model with a transition from heavier to lighter elements at these energies, with iron at the highest energies. This model produces a peak at exactly the right position, and therefore provides an especially good fit, but it contradicts the iron knee in the cosmic ray composition observed by the KASCADE experiment (Kampert et al. 2004). Note that all cases with a composition heavier than hydrogen at 100 TeV lead to a predicted neutrino flux about one order of magnitude below the flux required to describe the IceCube observation (IceCube Collaboration 2013).
We note that analytical estimates are not very accurate because: (a) the usual energy conservation arguments do not hold for spectra much steeper than E−2, (b) the cross-section increases with energy which induces a small spectral tilt, and (c) the distribution functions do have an impact.
3 RESULTS FOR THE TARGET DENSITY
The fluxes in Fig. 1 depend on the product Reff × nH. Here, we fit the computed neutrino spectra to the data in order to see what values can reproduce that, and what can be said about the fraction of neutrinos from cosmic ray interactions. We follow the method described in Winter (2013) updated by IceCube Collaboration (2013). The neutrino events detected by the IceCube detector are binned in four energy intervals 30–200 TeV, 0.2–1, 1–2, and 2–100 PeV. We use two different approaches. (1) Ignoring direction, we assume that all non-atmospheric events needs to be described by the interactions with hydrogen, computing the atmospheric background with the method in Winter (2013); model ‘All sky’. (2) We choose the events from the sky map (IceCube Collaboration 2013) which may potentially come from the cosmic ray interactions with the hydrogen halo within the directional uncertainties, and we correct for fraction of isotropically distributed events which may fall into the Galactic plane; model ‘Directional inf.’.1 The rest of the events are treated as (extragalactic and atmospheric) isotropic background. In addition, we assume that the neutrino directions are correlated with the diffuse gamma-ray emission from the Galactic plane, which is limited to a Galactic latitude below 5°, see Ackermann et al. (2012). This reduces the IceCube exposure to that flux by about a factor of 10 because of the reduced solid angle.
We present our main results in Table 2, where the best-fitting target densities and the χ2/d.o.f. are shown for different composition models (rows) and two different extreme models for the directional information and halo sizes (columns). Note that Reff = 10 kpc has been chosen for the ‘All sky’ model and Reff = 1 kpc for the directional model; for different values, the results can be easily re-scaled using Table 1. From the All sky model, only the pure hydrogen composition produces realistic values for nH, at the expense of a huge halo size.
. | All sky . | Directional inf. . | ||
---|---|---|---|---|
. | Reff = 10 kpc . | Reff = 1 kpc . | ||
Composition . | nH . | χ2 . | nH . | χ2 . |
. | [cm−3] . | /d.o.f. . | [cm−3] . | /d.o.f. . |
Hydrogen (A = 1) | |$1.6^{+0.3}_{-0.5}$| | 1.9 | |$72^{+48}_{-42}$| | 0.8 |
Helium (A = 4) | |$5.9^{+1.7}_{-1.5}$| | 2.1 | |$280^{+190}_{-170}$| | 0.8 |
Iron (A = 56) | |$130^{+38}_{-34}$| | 2.5 | |$6000^{+4300}_{-3800}$| | 0.9 |
Gaisser et al. (2013) | |$9.3^{+3.2}_{-2.8}$| | 5.1 | |$370^{+350}_{-300}$| | 1.3 |
Hypothetical | |$4.5^{+1.3}_{-1.2}$| | 1.4 | |$230^{+150}_{-130}$| | 0.7 |
. | All sky . | Directional inf. . | ||
---|---|---|---|---|
. | Reff = 10 kpc . | Reff = 1 kpc . | ||
Composition . | nH . | χ2 . | nH . | χ2 . |
. | [cm−3] . | /d.o.f. . | [cm−3] . | /d.o.f. . |
Hydrogen (A = 1) | |$1.6^{+0.3}_{-0.5}$| | 1.9 | |$72^{+48}_{-42}$| | 0.8 |
Helium (A = 4) | |$5.9^{+1.7}_{-1.5}$| | 2.1 | |$280^{+190}_{-170}$| | 0.8 |
Iron (A = 56) | |$130^{+38}_{-34}$| | 2.5 | |$6000^{+4300}_{-3800}$| | 0.9 |
Gaisser et al. (2013) | |$9.3^{+3.2}_{-2.8}$| | 5.1 | |$370^{+350}_{-300}$| | 1.3 |
Hypothetical | |$4.5^{+1.3}_{-1.2}$| | 1.4 | |$230^{+150}_{-130}$| | 0.7 |
. | All sky . | Directional inf. . | ||
---|---|---|---|---|
. | Reff = 10 kpc . | Reff = 1 kpc . | ||
Composition . | nH . | χ2 . | nH . | χ2 . |
. | [cm−3] . | /d.o.f. . | [cm−3] . | /d.o.f. . |
Hydrogen (A = 1) | |$1.6^{+0.3}_{-0.5}$| | 1.9 | |$72^{+48}_{-42}$| | 0.8 |
Helium (A = 4) | |$5.9^{+1.7}_{-1.5}$| | 2.1 | |$280^{+190}_{-170}$| | 0.8 |
Iron (A = 56) | |$130^{+38}_{-34}$| | 2.5 | |$6000^{+4300}_{-3800}$| | 0.9 |
Gaisser et al. (2013) | |$9.3^{+3.2}_{-2.8}$| | 5.1 | |$370^{+350}_{-300}$| | 1.3 |
Hypothetical | |$4.5^{+1.3}_{-1.2}$| | 1.4 | |$230^{+150}_{-130}$| | 0.7 |
. | All sky . | Directional inf. . | ||
---|---|---|---|---|
. | Reff = 10 kpc . | Reff = 1 kpc . | ||
Composition . | nH . | χ2 . | nH . | χ2 . |
. | [cm−3] . | /d.o.f. . | [cm−3] . | /d.o.f. . |
Hydrogen (A = 1) | |$1.6^{+0.3}_{-0.5}$| | 1.9 | |$72^{+48}_{-42}$| | 0.8 |
Helium (A = 4) | |$5.9^{+1.7}_{-1.5}$| | 2.1 | |$280^{+190}_{-170}$| | 0.8 |
Iron (A = 56) | |$130^{+38}_{-34}$| | 2.5 | |$6000^{+4300}_{-3800}$| | 0.9 |
Gaisser et al. (2013) | |$9.3^{+3.2}_{-2.8}$| | 5.1 | |$370^{+350}_{-300}$| | 1.3 |
Hypothetical | |$4.5^{+1.3}_{-1.2}$| | 1.4 | |$230^{+150}_{-130}$| | 0.7 |
Note that the statistics are good enough to derive lower bounds for the hydrogen density in the All sky case. In the directional model, the statistics are much poorer and the error bars therefore much larger. Because of the small solid angle coverage of the signal, the required target densities are extremely large, which is unlikely. However, the event rates in IceCube from the direction of the Galactic plane can be well reproduced, see Fig. 2. For the Gaisser et al. (2013) cosmic ray composition (left-hand panel), we obtain a relatively poor fit because of the dip at PeV (middle bins), exactly where the neutrino data require a peak (compare to Fig. 1). A better fit of the shape is, as expected, obtained for our hypothetical cosmic ray composition model, see right-hand panel. Although this model is incompatible with cosmic ray composition data, it may serve as a proof of principle that one can produce a peak at PeV with composition changes only. Note again that there is no direct dependence on the cosmic ray injection and escape time in our calculation.
We have calculated the secondary very high energy gamma-ray flux expected from π0 decays produced in pp interactions directly with Kelner et al. (2006). We show two different cases in Fig. 3: A = 1 All sky model versus Gaisser et al. composition model with directional information. Note that this result is shown for the best fit of the models to neutrino data, i.e. the normalization is determined by the neutrino observation and does not depend on nH or Reff individually. For illustration, we also show the curves for the gamma-ray fluxes corrected for absorption due to the background radiation with the mean free paths calculated in Protheroe & Biermann (1996) for d = 10 kpc. The upper limits on the diffuse gamma-ray flux from various experiments are compared with our results. One strong constraint comes from the KASCADE and CASA-MIA limits at a few hundred TeV. On the other hand, the Fermi-LAT observation at 100 GeV (Ackermann et al. 2012) does not impose a problem for the A = 1 ‘All sky’ model, whereas the directional model clearly exceeds the bound. The data above a few hundred TeV can be circumvented away by the attenuation of the gamma rays over long distances. The information given in Fig. 3 can be used to infer the fraction of neutrino events which can come from the interactions in our Galaxy by re-scaling the event rates to satisfy the bounds.
4 CONCLUSIONS
Taking into account the spectral shape of the observed neutrino spectrum, we have tested if it is plausible to describe the observed neutrino flux in the TeV–PeV range by interactions between cosmic rays and matter in the interstellar medium. We have discussed several composition models for the cosmic rays and several geometries for the target matter halo. For the directional information on the neutrino events, we have chosen two possibilities: either all events above the atmospheric backgrounds are to be described by the matter interactions or only the events compatible with the directions from the Galactic plane – whereas the rest forms an isotropic (atmospheric and extragalactic) background. In the latter case, we have also taken into account a probable correlation with the diffuse gamma-ray emission from the Galactic plane.
We have demonstrated that strong constraints arise from: (a) the expected target densities obtained from cosmic ray propagation models, (b) bounds on the diffuse gamma-ray emission from the Galactic plane, (c) the measured cosmic ray composition contradicting the flux shape observed in IceCube, and (d) the directional correlation with the diffuse gamma-ray emission from the Galactic plane, limiting the expected solid angle of the signal flux. In the most plausible scenario (directional information used, cosmic ray composition model by Gaisser et al. 2013), the required target density is about a factor of 100 above current expectations to describe the neutrino events from the direction of the Galactic plane. This means that only |$\mathcal {O}(0.1)$| event from the current IceCube observation would come from cosmic ray interactions for realistic target densities. Ignoring the directional information, a larger contribution |$\mathcal {O}(1)$| event is possible, taking into account the cosmic ray composition data, plausible halo sizes, and the gamma-ray constraints – which may serve as an upper limit for the estimate. However, this scenario requires unrealistically large target densities.
In conclusion, we have demonstrated that taking into account the known constraints, only a small fraction of the observed neutrino events may originate from the Galactic plane.
NG acknowledges local hospitality during her visit to Universität Würzburg. WW acknowledges support from DFG grants WI 2639/3-1 and WI 2639/4-1, the FP7 Invisibles network (Marie Curie Actions, PITN-GA-2011-289442), and the ‘Helmholtz Alliance for Astroparticle Physics HAP’, funded by the Initiative and Networking fund of the Helmholtz association. We are grateful to M. Ahlers for constructive comments.
We remove the events at the lowest energies for that, as expected for the atmospheric background, in the ratio 2:1 showers to tracks. That is, the remaining signal events are 2, 3, 4, 13, 14, 15, 22, 25, 27 following the numbering in table 1 of IceCube Collaboration (2013).